Martingale Method Research Articles

Optimal retirement planning under partial information

Abstract The present paper analyzes an optimal consumption and investment problem of a retiree with a constant relative risk aversion (CRRA) who faces parameter uncertainty about the financial market. We solve the optimization problem under partial information by making the market observationally complete and consequently applying the martingale method to obtain closed-form solutions to the optimal consumption and investment strategies. Further, we provide some comparative statics and numerical analyses to deeply understand the consumption and investment behavior under partial information. Bearing partial information has little impact on the optimal consumption level, but it makes retirees with an RRA smaller than one invest more riskily, while it makes retirees with an RRA larger than one invest more conservatively.

Size of the Largest Component in a Critical Graph

In this paper, by the branching process and the martingale method, we prove that the size of the largest component in the critical random intersection graphGn,n5/3,pis asymptotically of ordern2/3and the width of scaling window isn−1/3.

On the existence of semimartingales with continuous characteristics

We prove the existence of quasi-left continuous semimartingales with continuous local semimartingale characteristics which satisfy a Lyapunov-type or a linear growth condition, where latter takes the whole history of the paths into consideration. The proof is based on an approximation and a tightness argument and the martingale problem method.

PORTFOLIO OPTIMIZATION WITH PERFORMANCE RATIOS

We consider the portfolio selection problem of maximizing a performance measure in a continuous-time diffusion model. The performance measure is the ratio of the overperformance to the underperformance of a portfolio relative to a benchmark. Following a strategy from fractional programming, we analyze the problem by solving a family of related problems, where the objective functions are the numerator of the original problem minus the denominator multiplied by a penalty parameter. These auxiliary problems can be solved using the martingale method for stochastic control. The existence of solution is discussed in a general setting and explicit solutions are derived when both the reward and the penalty functions are power functions.

Deviation inequalities for separately Lipschitz functionals of composition of random functions

We consider a class of non-homogeneous Markov chains, that contains many natural examples. Next, using martingale methods, we establish some deviation and moment inequalities for separately Lipschitz functions of such a chain, under moment conditions on some dominating random variables.

Martingale and duality methods for optimal investment and reinsurance problem in a Lévy model

In this paper, we employ the martingale and duality methods to study the optimal investment and proportional reinsurance problem for an insurer. The insurer’s risk process is modeled by a Lévy process and the capital can be invested in a security market described by a geometric Lévy process. The objective of the insurer is to maximize the expected utility of her terminal wealth. We derive the expression for optimal investment-reinsurance strategies for various utility functions. Furthermore, an example is considered, and numerical simulations are presented to illustrate the effect of the parameters on the optimal strategies.

On Nodes of Small Degrees and Degree Profile in Preferential Dynamic Attachment Circuits

We investigate the joint distribution of nodes of small degrees and the degree profile in preferential dynamic attachment circuits. In particular, we study the joint asymptotic distribution of the number of the nodes of outdegree $0$ (terminal nodes) and outdegree $1$ in a very large circuit. The expectation and variance of the number of those two types of nodes are both asymptotically linear with respect to the age of the circuit. We show that the numbers of nodes of outdegree $0$ and $1$ asymptotically follow a two-dimensional Gaussian law via multivariate martingale methods. We also study the exact distribution of the degree of a node, as the circuit ages, via a series of P\'olya-Eggenberger urn models with "hiccups" in between. The exact expectation and variance of the degree of nodes are determined by recurrence methods. Phase transitions of these degrees are discussed briefly. This is an extension of the abstract [20].

Convergence of random bounded linear operators in the Skorokhod space

Abstract We introduce the space of random bounded linear operators on a separable Banach space such that their range belong to the Skorokhod space of right-continuous with left-hand limits functions. We call these random operators D-valued random variables. Almost sure and weak convergence results for the sequences of such random variables are proved by martingale methods. An application is described for a regime-switching inhomogeneous Lévy dynamics of a risky asset.

Portfolio choice, portfolio liquidation, and portfolio transition under drift uncertainty

This paper presents several models addressing optimal portfolio choice, optimal portfolio liquidation, and optimal portfolio transition issues, in which the expected returns of risky assets are unknown. Our approach is based on a coupling between Bayesian learning and dynamic programming techniques that leads to partial differential equations. It enables to recover the well-known results of Karatzas and Zhao in a framework a la Merton, but also to deal with cases where martingale methods are no longer available. In particular, we address optimal portfolio choice, portfolio liquidation, and portfolio transition problems in a framework a la Almgren–Chriss, and we build therefore a model in which the agent takes into account in his decision process both the liquidity of assets and the uncertainty with respect to their expected return.

Consumption–investment problem with pathwise ambiguity under logarithmic utility

For an investor with intertemporal information about risky assets, we propose a set of cadlag confidence paths to describe his ambiguity about drift, volatility, and jump of the risky assets. For each possible model, the differential characteristic of log-return processes is a stochastic process and almost surely takes value in the set of confidence paths. Under the framework of the robust consumption–investment problem for logarithmic utility, we prove that a worst-case confidence path exists and can generate a worst-case model and deduce an optimal strategy. A deterministic-to-stochastic paradigm is established and extends the classical martingale method to robust optimization for jump-diffusion model. In numerical analyses, we take a joint ambiguity with two-point jumps as an example to reveal the rule of choosing the worst-case model and the impact of the ambiguity on the optimal strategy.

Limit theorems for the fractional nonhomogeneous Poisson process

AbstractThe fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverseα-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.

Locality at the Boundary Implies Gap in the Bulk for 2D PEPS

Proving that the parent Hamiltonian of a Projected Entangled Pair State (PEPS) is gapped remains an important open problem. We take a step forward in solving this problem by showing two results: first, we identify an approximate factorization condition on the boundary state of rectangular subregions that is sufficient to prove that the parent Hamiltonian of the bulk 2D PEPS has a constant gap in the thermodynamic limit; second, we then show that Gibbs state of a local, finite-range Hamiltonian satisfy such condition. The proof applies to the case of injective and MPO-injective PEPS, employs the martingale method of nearly commuting projectors, and exploits a result of Araki on the robustness of one dimensional Gibbs states. Our result provides one of the first rigorous connections between boundary theories and dynamical properties in an interacting many body system.

Variance and Dimension Reduction Monte Carlo Method for Pricing European Multi-Asset Options with Stochastic Volatilities

Pricing multi-asset options has always been one of the key problems in financial engineering because of their high dimensionality and the low convergence rates of pricing algorithms. This paper studies a method to accelerate Monte Carlo (MC) simulations for pricing multi-asset options with stochastic volatilities. First, a conditional Monte Carlo (CMC) pricing formula is constructed to reduce the dimension and variance of the MC simulation. Then, an efficient martingale control variate (CV), based on the martingale representation theorem, is designed by selecting volatility parameters in the approximated option price for further variance reduction. Numerical tests illustrated the sensitivity of the CMC method to correlation coefficients and the effectiveness and robustness of our martingale CV method. The idea in this paper is also applicable for the valuation of other derivatives with stochastic volatility.

Optimal Stopping Time, Consumption, Labour, and Portfolio Decision for a Pension Scheme

In this work we solve in a closed form the problem of an agent who wants to optimise the inter-temporal utility of both his consumption and leisure by choosing: (i) the optimal inter-temporal consumption, (ii) the optimal inter-temporal labour supply, (iii) the optimal share of wealth to invest in a risky asset, and (iv) the optimal retirement age. The wage of the agent is assumed to be stochastic and correlated with the risky asset on the financial market. The problem is split into two sub-problems: the optimal consumption, labour, and portfolio problem is solved first, and then the optimal stopping time is approached. The martingale method is used for the first problem, and it allows to solve it for any value of the stopping time which is just considered as a stochastic variable. The problem of the agent is solved by assuming that after retirement he received a utility that is proportional to the remaining human capital. Finally, a numerical simulation is presented for showing the behaviour over time of the optimal solution.

VALUATION OF CONTINGENT GUARANTEES USING LEAST-SQUARES MONTE CARLO

AbstractWe consider the problem of pricing modern guarantee concepts in unit-linked life insurance, where the guaranteed amount grows contingent on the performance of an investment fund that acts simultaneously as the underlying security and the replicating portfolio. Using the Martingale Method, this nonstandard pricing problem can be transformed into a fixed-point problem, whose solution requires the evaluation of conditional expectations of highly path-dependent payoffs. By adapting the least-squares Monte Carlo method for American option pricing problems, we develop a new numerical approach to approximate the value of contingent guarantees and prove its convergence. Our valuation procedure can be applied to large-scale pricing problems, for which existing methods are infeasible, and leads to significant improvements in performance.

Finite-horizon optimal consumption and investment problem with a preference change

In this paper we propose a duality approach to solving an optimal consumption-portfolio selection problem in which an economic agent can choose (irreversible) the preference change time in finite horizon. We assume that after preference change, the agent's coefficient of relative risk aversion becomes higher than before. We use the martingale method and study the variational inequality or free boundary problem arising from the dual problem. Based on partial differential equation (PDE) theory, we analytically characterize the variational inequality and establish the duality relationship. Specifically, we prove that the agent will change his/her preference as soon as the agent's wealth reaches the threshold of wealth as a function of time.

Valuation of the Vulnerable Option Price Based on Mixed Fractional Brownian Motion

The pricing problem of a kind of European vulnerable option was studied. The mixed fractional Brownian motion and the jump process were used to characterize the evolution of stock prices. The closed-form solution to European option pricing was obtained by applying martingale measure transformation method. At the end of this paper, some numerical experiments were adopted to compare the new pricing formula introduced in this paper with the classical Black-Scholes pricing formula. The result showed that the new pricing formula conformed to the actual financial market. In fact, the option value is positively correlated with the underlying asset price and the company’s asset price and the jump process has significant influence on the value of option.

A class of stochastic Gronwall\u2019s inequality and its application

This paper puts forward the basic form of stochastic Gronwall’s inequality and uses, respectively, the iterative method, the integral method and the martingale representation method to prove it. Then it presents an application to prove a comparison theorem of L^{p} solutions for one-dimensional backward stochastic differential equations under the stochastic Lipschitz condition.

Optimal investment and benefit payment strategy under loss aversion for target benefit pension plans

In this paper, we consider the optimal investment and benefit payment strategy for a target benefit plan (TBP), where the plan members are loss averse with an S-shaped utility over benefit relative to a time-varying target benefit level. The pension payments are dependent on the financial situation of the plan, with risk sharing between different generations. The pension fund is invested in both a risk-free asset and multiple risky assets. Using the martingale method, we derive the optimal investment strategy and optimal benefit payment policy, explicitly, which minimizes the interim utility of the benefit risk in terms of deviating from the benefit target. Finally, some numerical examples and sensitivity analyses are provided to show the effects of market parameters on the optimal strategies. We also compare the optimal benefit payment policy for loss-averse participants with that of constant relative risk averse (CRRA) participants by numerical results. We find that the TBP model for loss-averse participants is effective in providing a stable and sustainable pension account for participants.

Pricing cross-currency interest rate swaps under the Levy market model

This paper derives a pricing model for interest rate swaps when the underlying markets and settlement currency can be set arbitrarily. Using the risk-neutral valuation method developed by Musiela and Rutkowski (Martingale methods in financing modelling, 2nd edn, Springer, New York, 2005), the authors generate arbitrage-free prices for a Levy market. The Levy processes are attractive because they support better statistical fits than a Gaussian economy. A closed-form solution for the swap value results from replicating the payment at each settlement date. The results then show that the domestic and foreign term structures are important factors in the pricing model; the swap value contains a correction term that reflects the currency hedging cost for the correlation between interest rates and the exchange rate.